doubly transitive groups are primitive

Theorem.

Proof.

Let G acting on X be doubly transitive. To show the action is , we must show that all blocks are trivial blocks; to do this, it suffices to show that any block containing more than one element is all of X. So choose a block Y with two distinct elements y1,y2. Given an arbitrary x∈X, since G is doubly transitive, we can choose σ∈G such that

σ⋅(y1,y2)=(y1,x)

But then σ⋅Y∩Y≠∅, since y1 is in both. Thus σ⋅Y=Y, so x∈Y as well. So Y=X and we are done.
∎